[Dire]

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against big hands a small bet is good because we'll look like we are vb AcXc and he'll raise.

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I think he never check raises worse on this river, regardless of how much we bet. Imagine he has AJ. He's still not check raising a $40 bet. He likely feels he has alot of our range beat, but he's completely dead to our check raise calling range on this board. And I doubt he decides to try to pull of some crazy bluff here.

Same reason I just like betting big on this river, preferably shoving. He's only calling with big hands, and people don't like to fold big hands to odd lines.